Section: New Results
Interacting Markov chain Monte Carlo methods
Participants : Bernard Bercu, Anthony Brockwell, Pierre Del Moral, Arnaud Doucet.
This new line of research is mainly concerned with the design and the analysis of a new class of interacting stochastic algorithms for sampling complex distributions including Boltzmann-Gibbs measures and Feynman-Kac path integral semigroups arising in physics, in biology and in advanced stochastic engineering science. These interacting sampling methods can be described as adaptive and dynamic simulation algorithms which take advantage of the information carried by the past history to increase the quality of the next series of samples. One critical aspect of this technique as opposed to standard Markov chain Monte Carlo methods is that it provides a natural adaptation and reinforced learning strategy of the physical or engineering evolution equation at hand. This type of reinforcement with the past is observed frequently in nature and society, where beneficial interactions with the past history tend to be repeated. Moreover, in contrast to more traditional mean field type particle models and related sequential Monte Carlo techniques, these stochastic algorithms can increase the precision and performance of the numerical approximations iteratively. The origins of these interacting sampling methods can be traced back to a pair of articles [35] , [36] by P. Del Moral and L. Miclo. These studies are concerned with biology-inspired self-interacting Markov chain models with applications to genetic type algorithms involving a competition between the natural reinforcement mechanisms and the potential attraction of a given exploration landscape.
In 2008-2009, these lines of research have been developed in three different directions :
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In [4] , we introduce a novel methodology for sampling from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. These problems are usually addressed using Sequential Monte Carlo (SMC) methods. The alternative Sequentially Interacting Markov Chain Monte Carlo (SIMCMC) scheme proposed here works by generating interacting non-Markovian sequences which behave asymptotically like independent Metropolis-Hastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC methods, this scheme allows us to iteratively improve our estimates in an MCMC-like fashion. We establish convergence of the algorithm under realistic verifiable assumptions and demonstrate its performance on several examples arising in Bayesian time series analysis.
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In [5] , We present a new class of interacting Markov chain Monte Carlo algorithms for solving numerically discrete-time measure-valued equations. The associated stochastic processes belong to the class of self-interacting Markov chains. In contrast to traditional Markov chains, their time evolutions depend on the occupation measure of their past values. This general methodology allows us to provide a natural way to sample from a sequence of target probability measures of increasing complexity. We develop an original theoretical analysis to analyze the behaviour of these iterative algorithms which relies on measure-valued processes and semigroup techniques. We establish a variety of convergence results including exponential estimates and a uniform convergence theorem with respect to the number of target distributions. We also illustrate these algorithms in the context of Feynman-Kac distribution flows.
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In [3] , we present a functional central limit theorem for a new class of interaction Markov chain Monte Carlo interpretations of discrete generation measure valued equations. We provide an original stochastic analysis based on semigroup techniques on distribution spaces and fluctuation theorems for self-interaction random fields. Besides the fluctuation analysis of these models, we also present a series of sharp
-mean error bounds in terms of the semigroup associated with the first order expansion of the limiting measure valued process, yielding what seems to be the first results of this type for this class of interacting processes. We illustrate these results in the context of Feynman-Kac integration semigroups arising in physics, biology and stochastic engineering science.
